What attributes other than magnitude and sign distinguish one real number from another?

Can it be proved that small numbers have a greater density of "mathematical characteristics" (e.g., fundamental constants, general members of sequences, overall usage, etc.) than relatively large numbers do?

If I remember correctly, one of my profs mentioned that the real numbers were defined as the set of all numbers that are converged to by some rational cauchy sequence, so I'm not sure if any one number has more sequences than the rest. In terms of usage, I would guess $0,1,2$ to be used the most, since they each have fundamental properties in $\mathbb{R}$.